Basins of attraction in strongly damped coupled mechanical oscillators: A global example

0044-2275/99/020282-19 $ 1.50+0.20/0

c 1999 Birkh¨ auser Verlag, Basel Mathematik und Physik ZAMP

Basins of attraction in strongly damped coupled mechanical oscillators: A global example ^∗

Bernold Fiedler, Mohamed Belhaq and Mohamed Houssni

Abstract. We consider a finite array of N oscillators with nearest neighbor coupling along a line, and with two types of damping. Friction terms can affect each individual oscillator, separately:

local damping. Neighboring damping, in contrast, affects nearest neighbor distances.

Although stability of equilibria does not depend on the particular type of damping, global basins of attraction do. We show that basins of attraction can in fact jump discontinuously under continuous variations of local versus neighbor damping. This effect is caused by heteroclinic saddle-saddle connections of equilibria. It occurs even in the limit of strong damping and for only two oscillators, N = 2.

The results are based on geometric singular perturbation methods, Sturm type oscillation theory (zero numbers), and the related theory of Jacobi systems. Going beyond the motivating mechanical application, they emphasize the dependence of basins of attraction and heteroclinic orbit connections in gradient systems on the underlying metric.

Mathematics Subject Classification (1991). 34C37, 70K20, 34D15, 34D45.

Keywords. Coupled oscillators, basins of attraction, gradient systems, heteroclinic connections, Jacobi systems, zero number, lattice dynamics.

1. Introduction

A mechanical oscillator with one degree of freedom and with a linear friction force can be modelled as a pendulum

¨

u + α u ˙ + f (u) = 0. (1.1)

In the absence of damping, that is for α = 0, a strict local minimum u 0 of the potential energy F(u) = R

f (s)ds is surrounded by periodic orbits in the phase plane (u, u) ˙ ∈ X = R ² . For positive damping, α > 0, the minimum u 0 becomes an attracting equilibrium (u 0 , u ˙ ≡ 0) of (1.1) in the phase plane X. Its basin of

∗

This work was supported by the Deutsche Forschungsgemeinschaft and by the Deutsche

Gesellschaft f¨ ur Technische Zusammenarbeit.

attraction, B(u 0 ), consists of all initial conditions (u(0), u(0)) in ˙ X such that the solution of (1.1) satisfies

t→

lim +

(u(t), u(t)) = (u ˙ ₀ , 0). (1.2) The basin of attraction depends on α and can, in fact, undergo quite intricate global changes as the Hamiltonian situation α = 0 is approached by the gradient- like cases α > 0 with Lyapunov function ˙ u ² /2 + F (u); see [Ter85].

In the limit of strong damping, α → + ∞ , system (1.1) reduces to the gradient flow

˙

u = − f (u). (1.3)

Indeed, we may rescale time (t → αt) and rewrite (1.1) as a system u

= v

α

² v

= − v − f (u) (1.4)

Standard geometric singular perturbation theory ([Fen79], Wig94]) then identifies a slow manifold

v = − f (u) + O (α

² ). (1.5)

in the phase plane (u, v) ∈ X which is normally hyperbolic with large exponential rate ∼ α ² of attraction. Inside this “inertial manifold”, being a graph over u, the flow is given by

u

= − f (u) + O (α

² ). (1.6)

The perturbation is small of the order indicated, uniformly for derivatives up to the order of differentiability of the force f (u). This justifies the limit (1.3).

A finite array of N mechanical oscillators, linearly coupled along a line with damping coefficients α

, β

and coupling strengths λ

, can be modelled by

0 = ¨ u

+ α

u ˙

+ f

(u

) + β

1 ( − u ˙

1 + ˙ u

) + β

( − u ˙

+1 + ˙ u

)+

+ λ

1 ( − u

1 + u

) + λ

( − u

+1 + u

), (1.7) where i = 1, . . . , N . Neumann boundary conditions amount to

u 0 := u 1 , u

+1 := u

. (1.8) Dirichlet conditions, for example, would be u ₀ := 0, u

+1 := 0.

Note that (1.7) can also be viewed as a spatial finite difference semidiscretiza- tion of a hyperbolic partial differential equation

0 = u

+ αu

+ f (x, u) − (βu

)

− (λu

)

We call the effect of α

> 0 a local damping, whereas β

> 0 indicate neighbor damping. Rewriting (1.7), (1.8) in vector form u = (u 1 , . . . , u

), we obtain

0 = ¨ u + αA u ˙ + Lu + f (u) (1.9) We henceforth assume α > 0, L positive semidefinite, A strictly positive definite, and A, L symmetric. This is the case for λ

> 0, α

> 0, β

≥ 0. Note how the boundary conditions enter into L. Also note that (f (u))

= f

(u

).

In the following, we fix A, L and consider the limit of strong damping, α → + ∞ . By the same arguments as in (1.3)-(1.6) above, the system 1.9 reduces to

A u ˙ = − f (u) − Lu (1.10)

on a slow manifold v = − f (u) − Lu, for α = + ∞ . The behavior of basins of attraction for α ≥ α ₀ large, with (1.9) limiting to (1.10), will be our main object of study, in the present paper. In particular theorem 1.1 below investigates discontinuous jumps in basin boundaries as the damping is varied from local to neighbor type.

System (1.10) is gradient-like with respect to the potential (or energy or Lya- punov) function

V (u) := F ₁ (u ₁ ) + . . . + F

(u

) + 1

2 u

Lu, (1.11)

where F

denote the primitives of f

, as before. Indeed, (1.10) reads

A u ˙ = − grad

V (u) (1.12)

and therefore positivity of A implies d

dt V (u) = − (grad

V (u))

A

¹ grad

V (u) < 0 (1.13) is strictly negative along solutions u(t) of (1.10), unless grad

V (u) = 0. In partic- ular, bounded solutions u(t) tend to the set of equilibria for t → + ∞ , t → −∞ , respectively.

Consider, more specifically, an equilibrium u = E 0 which is asymptotically stable, that is, a strict local minimum of the potential V (u). Note that E 0 remains an equilibrium, for all choices of the positive definite symmetric damping matrix A. In section 2, we prove that (linear) stability of E ₀ is also independent of A.

Therefore it makes sense to define its basin of attraction, B(E ₀ ), as in (1.2) above.

Our main result, theorem 1 below, states that the basin of attraction B(E ₀ ) can

depend on the precise form of the damping matrix A, even in the limit of large

damping α = + ∞ which we consider in (1.10).

To be completely specific, we consider (1.7) for the following example of N = 2 oscillators

f ₁ (u ₁ ) = − 1 2 u ³ ₁ − 3

2 u ² ₁ − u ₁ + 1 f 2 (u 2 ) = − u ² ₂

α ₁ = α ₂ = (1 − τ)α (1.14)

β 1 = τ α λ 1 = λ

under Neumann boundary conditions (1.8) and in the limit α → + ∞ of large damping. Here λ > 0, and the homotopy parameter 0 ≤ τ < 1 indicates the relative size of the two types of strong damping: τ = 0 indicates purely local damping and absence of neighbor friction, whereas τ % 1 denotes the limit of pure neighbor damping of individually undamped mechanical oscillators. We now state our main result.

Theorem 1.1. Consider the four-dimensional system (1.7), (1.8) of coupled me- chanical oscillators with N = 2 degrees of freedom and with specific nonlinearities and parameters (1.14). For 0 ≤ τ < 1, α > 0 and all λ sufficiently close to 1, the following then holds.

System (1.7), (1.8) possesses precisely four equilibria E 0 , E 1

, E 2 . Each equi- librium is hyperbolic, that is, an isolated, nondegenerate critical point of the poten- tial V (u) defined in (1.11). The unstable dimensions, alias Morse indices, of these four equilibria are 0, 1, 1, 2, respectively, as indicated by their numerical subscripts.

In particular, E ₀ is attracting with basin B(E ₀ ).

For strong damping, that is, for some sufficiently large α ₀ and any fixed α ≥ α ₀ > 0, the basin B(E ₀ ) depends significantly on the relative size τ ∈ [0, 1) of neighbor versus local damping.

More precisely, there exist parameters λ ₀ near 1 and τ(α) ∈ (0, 1), such that for λ = λ ₀ and all τ ∈ (0, 1) \ τ(α) sufficiently close to τ(α) the following dichotomy arises. For τ on one side of τ(α), the repelling equilibrium E ₂ lies in the boundary

∂B(E 0 ) of the E 0 -basin, whereas E 2 6∈ ∂B(E 0 ) for τ on the other side of τ(α).

In particular, the basin boundary experiences a finite nonzero jump as the rel- ative damping τ increases through τ = τ(α). The jump size is bounded away from zero, uniformly for all α ≥ α 0 . This effect is caused by a nongeneric, nontrans- verse saddle-saddle heteroclinic orbit from E 1+ to E 1

at τ = τ(α).

We emphasize that our results are proved with mathematical rigor rather than just suggested by numerical simulation. For illustration of the geometry in the planar limiting system (1.10) of damping α = + ∞ , we refer to Figs. 5.1.a,b. For a general background of methodology we refer to [Fen79], [CH82], [GH83], [Wig94].

Altough the explicit statement of our theorem is planar, we emphasize that

our methodology is not planar. In principle, it applies to any number of coupled

oscillators. Our approach will carefully outline a general strategy to detect jumps in basin boundaries induced by a transition from local to neighbor friction. In this sense, our specific planar example serves as a minimalistic paradigm for the general situation.

Outline of the paper. In section 2, we investigate equilibria and the variational structure of the limiting system (1.10) at α = + ∞ , including issues like Morse indices, heteroclinic connections, and boundaries of basins of attraction. Section 3 is devoted to nodal properties of (1.10). This is a global structure of Sturm oscillation type, which is peculiar to the case τ = 0 of purely local damping.

In contrast to theorem 1.1, systems (1.10) with purely local damping are Morse- Smale, if all equilibria are hyperbolic. In particular, saddle-saddle heteroclinics as in theorem 1.1 cannot occur for τ = 0, see proposition 3.1.

In section 4, we address the limit τ % 1 by methods of geometric singular perturbation theory. In particular, the degenerate saddle-saddle heteroclinic orbit from E 1+ to E 1

will appear at coupling parameter λ = 1, in this limit. Through- out sections 2-4 we will develop our approach to coupled mechanical oscillators in complete generality, not just restricted to our particular example (1.14). Only in section 5, we condense our approach into a rigorous proof of theorem 1. An out- line of the proof, which also summarizes sections 2–4 with a view towards example (1.14), is given at the beginning of section 5. We conclude, in section 6, with a discussion of our results.

2. Variational structure

In the introduction we have seen how variational gradient systems (1.10) arise from coupled mechanical oscillators (1.7), (1.8) in the limit of strong damping. We now collect some facts on gradient systems, in their own right, for later use. We first observe that the unstable dimension (= Morse index) of equilibria does not depend on the type τ of damping, local versus neighbor; see proposition 2.1. Next, we give a brief account of structural stability of Morse-Smale systems, as introduced by Palis and Smale [PS70]; see also [PdM82]. We conclude, in proposition 2.3, with an investigation of isolated equilibria in the basin boundary of attractors.

To address hyperbolicity of equilibria, we consider the following slight general- ization of (1.10), (1.12):

A(u) ˙ u = − grad

V (u) =: − V

(u) (2.1) where A(u) is a C ¹ function of strictly positive definite, symmetric matrices, and V is a C ² scalar function of u ∈ R

. Note that (2.1) is gradient-like with respect to the Lyapunov functional V . For hyperbolic equilibria, alias critical points u = E of V , we let i(E) denote the unstable dimension, alias Morse index of E, that is, the number of eigenvalues of the linearization

− A(E)

¹ V

(E) (2.2)

with strictly positive real part, counting algebraic multiplicity.

Proposition 2.1. The Morse index i(E) of the equilibrium E of (2.1) does not depend on the matrix function A(u). In particular, E is hyperbolic if, and only if, it is nondegenerate as a critical point of V . The unstable dimension of E, in (2.2), equals the Morse index of E, as a critical point of V .

Proof. By linear conjugation with the positive definite, symmetric square root A(E) ¹

² , the matrix (2.2) transforms into

− A(E)

¹

² V

(E)A(E)

¹

² . (2.3) As a quadratic form, this latter matrix is equivalent to the Hessian − V

(E) itself.

In particular, the Morse indices coincide, and the proof is complete.

So, the proof is by basic linear algebra. A slightly more abstract view point would observe that the notion of a gradient grad

V (u) depends on the underlying Riemannian metric and the associated direction − grad

V (u) of steepest descent.

Thus, the metric A(u) does not change the Morse index, or saddle type, of the nondegenerate critical point u = E. Nevertheless, a particular change in the damping metric A(τ) could change the global pattern of basins and heteroclinic orbits, in our context of coupled mechanical oscillators, at least in principle. Our main result, theorem 1.1, states that this actually happens in our specific example (1.14).

We now begin our investigation of heteroclinic orbits and basins of attraction of general gradient-like systems

˙

u = g(u), (2.4)

u ∈ R

. By gradient-like we mean that there exists a Lyapunov function V such that t → V (u(t)) decreases strictly along solutions u(t) except, of course, at equilibria, where g(E) = 0. Note that (1.10), (1.12) and, more generally, (2.1) are gradient-like systems. Similarly, the original system of coupled oscillators (1.7), (1.8) is gradient-like, provided all damping coefficients α

are positive (but not necessarily large).

We recall that α- and ω-limit sets of bounded solutions u(t) consist entirely of equilibria, for gradient-like systems, by LaSalle’s invariance principle; see [Hal69].

If equilibria are isolated, for example hyperbolic as they will be in our context, then α- and ω-limit sets consist of single equilibria, respectively. Denote the (local) flow of (2.4) by u(t) = u 0 · t; the initial condition is u(0) = u 0 . We call an equilibrium E 0 an attractor, if

t→

lim +

u 0 · t = E 0 (2.5)

for all initial conditions in a sufficiently small neighborhood of E ₀ . In particular,

E ₀ is a strict local minimum of V , and hence stable, and is isolated as an equi-

librium. Its basin of attraction B(E ₀ ) is the set of all u ₀ , including E ₀ , for which

(2.5) holds. Attractors have open basins. By ∂B(E 0 ) we denote the topological basin boundary. Both B and ∂B are flow invariant, in both positive and negative time direction.

We say that an equilibrium E 2 connects to another equilibrium E 1 by a hete- roclinic orbit u ₀ · t, symbolically E ₂ E ₁ , if

t→−∞

lim u ₀ · t = E ₂

t→

lim +

u 0 · t = E 1

(2.6)

If both E ₁ and E ₂ are hyperbolic, then (2.6) is equivalent to a nonempty intersec- tion of the respective stable and unstable manifolds:

u 0 ∈ W

(E 2 ) ∩ W

(E 1 ). (2.7) We call a gradient-like system Morse-Smale, if all equilibria are hyperbolic and the intersections (2.7) are all transverse. Structural stability holds for Morse-Smale systems on compact manifolds: any C ¹ -close gradient-like flow can by conjugated to the original, unperturbed flow by a homeomorphism h which preserves time orbits and is C ⁰ -close to identity [PS70], [PdM82]. In addition to this openness property, the Morse-Smale property is generic, in particular dense, in the class of all gradient-like flows. Similar results hold for gradient-like system on R

, replacing the compact manifold, if we require dissipativity: the set of equilibria is uniformly bounded, and

V (u) → + ∞ , for | u | → ∞ . (2.8) To see this just compactify the flow to a Morse-Smale flow on S

, with a repelling north pole representing infinity.

Proposition 2.2. Consider a Morse-Smale system on a compact manifold or, alternatively, on R

with dissipativity. Let E 0 be an attractor, and hence of zero Morse index i(E 0 ), with basin B(E 0 ) and basin boundary ∂B(E 0 ).

Then B(E 0 ), ∂B(E 0 ) change continuously, with respect to symmetric Hausdorff distance, under C ¹ gradient-like, dissipative perturbations. Continuity is under- stood as uniform convergence on bounded subsets.

More precisely, the basin boundary ∂B(E ₀ ) is the union of stable manifolds W

(E) of all equilibria E which connect heteroclinically to E ₀ :

∂B(E 0 ) = [

E E₀

W

(E). (2.9)

In particular, an equilibrium E lies in the basin boundary ∂B(E ₀ ) if, and only if,

E E ₀ .

Proof. Continuity of B(E 0 ) = W

(E 0 ) and ∂B(E 0 ) follow from the work of Smale and Palis. They also show that the relation E E

is transitive, for Morse- Smale systems. Now let u 0 ∈ W

(E) for some E E 0 . Then their arguments also yield u 0 ∈ ∂B(E 0 ). Therefore ∂B(E 0 ) contains the right hand side of (2.9).

Conversely, let u ₀ ∈ ∂B(E ₀ ) be a limit of u

₀ ∈ B(E ₀ ) = W

(E ₀ ). Passing to subsequences, if necessary, the trajectories u

₀ · t, t ≥ 0, then converge to a finite sequence of equilibria E

, . . . , E ₁ in ∂B(E ₀ ) of strictly decreasing Morse index such that

u ₀ · t E

, for t → + ∞ , (2.10)

and to additional heteroclinic orbits

E

. . . E ₁ E ₀ . (2.11)

In particular, u 0 ∈ W

(E

) and, by transitivity of , also E

E 0 . Defining E = E

completes the proof of (2.9) and of the proposition.

We remark that proposition 2.3 adapts to nondissipative Morse-Smale systems, like example (1.14), as follows. Basin boundaries ∂B(E 0 ) still contain the right hand side of (2.9). If B(E ₀ ) is unbounded, then ∂B(E ₀ ) may contain additional orbits which are unbounded in forward time, or which connect to equilibria which themselves connect to infinity. However, a change in the set of equilibria E which connect to E ₀ still implies a jump in the basin boundary. Indeed these claims can easily be verified by a standard cut-off procedure which renders the system dissipative.

3. Local damping and Sturm properties

In this section we consider gradient systems

α ⁰

u ˙

= − f

(u

) + λ

1 (u

1 − u

) + λ

(u

+1 − u

), (3.1) i = 1, ..., N, with Neumann boundary conditions u ₀ := u ₁ , u

+1 := u

and positive coupling constants λ

as well as damping rates α ⁰

> 0. This corresponds to large, purely local damping with zero neighbor damping, β

= 0. The system is of form (1.10) with diagonal damping matrix A. Abstractly (3.1) can be rewritten as

˙

u

= g

(u

1 , u

, u

+1 ). (3.2) The tridiagonal nonlinearity g

has positive partial derivatives with respect to the off-diagonal entries u

1 , because

∂

1 g

= λ

1 > 0

∂

+1 g

= λ

> 0 (3.3)

In other words, (3.2) is a Jacobi system in the sense of Fusco and Oliva, see [FO88].

Jacobi systems possess a characteristic nodal property or Sturm property. For any vector η = (η 1 , ..., η

) ∈ R

let z(η), the zero number of η, denote the number of strict sign changes of the ordered sequence η 1 , ..., η

. Now consider any two solutions u ¹ (t), u ² (t) of a Jacobi system (3.2). Then

t 7→ z(u ² (t) − u ¹ (t)) (3.4)

is nonincreasing with t. More precisely, z drops strictly at t = t ₀ whenever the difference η(t ₀ ) := u ² (t ₀ ) − u ¹ (t ₀ ) ∈ R

possesses a “multiple zero”:

η

(t 0 ) = 0 and η

1 (t 0 ) · η

+1 (t 0 ) ≥ 0.

(We use Neumann boundary conditions η ₀ = η ₁ , η

+1 = η

at i = 1, N, here.) For linear parabolic boundary value problems on the interval, this structure was discovered by Sturm [Stu36] in 1836. For nonlinear PDEs, the subject was suc- cessfully revived by [Mat82]. For Jacobi systems, see [FO88].

Proposition 3.1. [FO88] Consider a Jacobi system (3.2) with a finite number of equilibria. If all equilibria are hyperbolic, then the Jacobi system is Morse-Smale.

We note that an explicit Lyapunov function for Jacobi systems was constructed in [FG97], identifying (3.2) to be gradient-like in general. In our special oscillator case (3.1), a gradient-like structure is inherited from the potential energy V, of course; see (1.11).

Fusco and Oliva prove the Morse-Smale property of Jacobi systems, in [FO88], by establishing the one missing ingredient: transversality of stable and unstable manifolds. The Sturm property (3.4) is the crucial tool in their proof. Note that the Morse-Smale property excludes, in particular, heteroclinic connections E E

between hyperbolic equilibria of equal Morse index i(E) = i(E

).

For dissipative Morse-Smale systems (3.2), Fiedler and Rocha have developed an explicit algorithm to determine which equilibria possess a heteroclinic connection, and which do not. The only required input information is the relative ordering of all equilibria at the left boundary, i = 1, versus the right boundary, i = N. See [FR96a], [FR96b] for details. In particular, the algorithm explicitly identifies all equilibria E which connect to a given attractor E 0 , in our description (2.9) of the basin boundary ∂B(E 0 ).

Our specific example (1.14) is not dissipative. It is possible to adapt the theory in [FR96a] to cover that case, confirming the numerical result in Fig. 6.1 below.

Since this adaptation will not be used in our proof of theorem 1.1, though, we do not dwell on the necessary details here.

Because the Morse-Smale property is open (see section 2), the system (1.7) of coupled mechanical oscillators with β

= 0 becomes Morse-Smale for α

= α ⁰

· α, α ⁰

> 0, and

α ≥ α ⁰ (α ⁰ ₁ , ..., α ⁰

) (3.5)

Indeed, the bounded solutions are contained in the strongly attracting slow man- ifold, where the flow limits onto (3.1); see (1.9), (1.10), and section 4. Within the slow manifold, the Morse-Smale argument applies.

In particular, we conclude that saddle-saddle connections E E

, i(E) = i(E

) are impossible in the case of strong damping of purely local type.

4. Neighbor damping and singular perturbations

In contrast to the previous section, we now consider the limit of strong damping of pure neighbor type. Specifically, as in (1.9), (1.10), we consider systems

A(τ) ˙ u = − f (u) − Lu (4.1) with diagonal nonlinearity (f (u))

= f

(u

) and linear nearest neighbor coupling L. The damping matrix A(τ) corresponds to choices

α

= (1 − τ)α · α ⁰

β

= τ α · β ⁰

(4.2) 0 ≤ τ < 1, in the limit α → + ∞ of strong damping. In complete detail, A(τ) = (1 − τ)A 0 + τ A 1 with symmetric damping matrices

A 0 =



  α ⁰ ₁

. ..

α ⁰



  , A 1 =



 

 



β ₁ ⁰ − β ₁ ⁰

− β ₁ ⁰ (β ₁ ⁰ + β ⁰ ₂ ) . ..

. .. . .. − β

⁰

− β

⁰ β

⁰



 

 

 . (4.3)

In accordance with section 1, we assume α ⁰

> 0, β ⁰

> 0, for i = 1, ..., N. Note that the local damping matrix A 0 is strictly positive definite, whereas the neighbor damping matrix A 1 is only positive semidefinite with one-dimensional kernel given by e = (1, ..., 1) :

ker A ₁ = span e = co-ker A ₁ (4.4)

By standard perturbation theory [Kat80], the orthogonal eigenprojection P 0 = N

¹ ee

associated to the simple eigenvalue ε = 0 of the symmetric matrix A 1 = A(τ = 1) continues differentiably to a projection onto the eigenspace of the corres- ponding eigenvalue ε near zero of A(τ ), for τ ∈ [0, 1) near 1. Note the expansion

ε = (1 − τ ) · ( 1 N

X

=1

α ⁰

) + o(1 − τ). (4.5)

This allows us to introduce the eigenvalue ε > 0 as a new, small parameter, replacing τ near 1. With eigenprojections P

, Q

:= 1 − P

, and the associated decomposition

u = v + w, v := P

u (4.6)

we obtain the singular perturbation form

ε v ˙ = − P

(f (v + w) + L(v + w))

˙

w = − A

(ε)Q

(f (v + w) + L(v + w)), (4.7) Here A

(ε) denotes the inverse of A(τ) on the invariant subspace given by range Q

. Note that A

(ε = 0) is the usual pseudo inverse A

₁ of A ₁ . In fast time t

= t/ε, we obtain the reduced fast system

v

= − ( 1 N

X

=1

f

(v

+ w

)) · e w

= 0

(4.8)

in the limit ε & 0. Here

= d/dt

, and we have used P ₀ L = 0. In slow time t, we obtain the corresponding slow system

0 = − ( 1 N

X

=1

f

(v

+ w

)) · e

˙

w = − A

₁ Q ₀ (f (v + w) + Lw),

(4.9)

formally, for ε & 0. We have used Q ₀ Lv = LQ ₀ v = 0 here. Finally we let S ₀ := { u ∈ R

|

X

=1

f

(u

) = 0 } (4.10)

denote the (limiting) slow manifold. For S ₀ to actually be a manifold we require 0 to be a regular value of the function

u 7→

X

=1

f

(u

). (4.11)

In other words, P

f

(u

) = 0 implies f

(u

) 6 = 0, for some i. The slow manifold is the equilibrium set of the fast system (4.8) or, alternatively, the set where the slow system (4.9) makes sense. The manifold S 0 can be written as a graph

S ₀ : v = ψ ₀ (w) (4.12)

locally, by the implicit function theorem, near points u ∈ S 0 where X

i

=1

f

(u

) 6 = 0. (4.13)

We call these u regular points of S ₀ ; all other points of S ₀ are called singular.

Proposition 4.1. In compact regions of regular points, bounded away from the singular set, the slow manifold S ₀ is uniformly normally hyperbolic under the fast flow (4.8). The slow manifold is exponentially attracting, if

X

=1

f

(u

) > 0, (4.14)

and exponentially repelling in case of negative sign. In either case, S ₀ continues to a (nonunique) normally hyperbolic invariant manifold S

= graph ψ

, for 0 < ε ≤ ε ₀ , uniformly in the compact region.

Similarly, let E be a hyperbolic equilibrium of (4.1) at a regular point of S ₀ . Then the Morse index i

(E) of E within the slow flow on S

, where v = ψ

(w), coincides with the ε-independent Morse index i(E) in the full system (4.7), for 0 < ε ≤ ε 0 , in the attracting case (4.14). Moreover, the local unstable manifolds W

(E) converge to the local unstable manifold W ₀

(E) of E for the slow flow (4.9) within S 0 , in the C ¹ -topology. The local stable manifolds W

(E), however, converge in C ¹ to the product

W ₀

(E) ⊕ span _loc { e } , (4.15) where span _loc denotes a local span and W ₀

(E) is again understood for the slow flow (4.9) within S ₀ .

In the repelling case of a negative sign in (4.14), the roles of the stable and unstable manifolds are reversed, and in particular i(E) = i

(E) + 1.

For a proof we refer to geometric singular perturbation theory [Fen79]; see also [Wig94]. The appearance of ε in (4.7) together with regularity condition (4.13) ensure, in fact, normal hyperbolicity in the sense of [HPS77] with exponential normal contraction/expansion rate of the order O (1/ε).

5. Proof of theorem 1.1

We give the proof, postponing three computational details. We postpone show-

ing that the system (1.7), (1.8) of N = 2 coupled mechanical oscillators with

nonlinearities and parameters (1.14) possesses precisely four equilibria E ₀ , E ₁

, E ₂

of the appropriate Morse indices, for coupling constant λ = 1. See (i) below. By proposition 2.1, the Morse indices do not depend on the damping coefficients α > 0, 0 ≤ τ < 1. Hyperbolicity persists for λ near 1, independently of α, τ . For τ = 0, α = + ∞ , the system is Morse-Smale, by proposition 3.1, independently of λ, as long as hyperbolicity of equilibria persists. This structure persists for large damping α ≥ α ⁰ ; see (3.5).

Still for α = + ∞ , but in the singular limit τ = 1, alias ε = 0, we obtain a degenerate formal saddle-saddle connection E ₁₊ E ₁

. As λ increases through 1, the (formal) manifolds W ₀

(E ₁₊ ), W ₀

(E ₁

) cross each other with nonvanishing speed, that is, transversely with respect to λ, as we postpone showing; see (ii) below. By proposition 4.1, this transversality feature persists, for τ % 1, at a unique point

λ = λ(τ) (5.1)

near λ(1) = 1. By analyticity of separatrices, the continuous function λ(τ) is analytic for τ < 1. Note that λ(τ) 6≡ 1, because a saddle-saddle connection is impossible at τ = 0, due to the Sturm structure there. Now fix values λ 0 , τ 0 near 1 such that λ(τ 0 ) = λ 0 and λ

(τ 0 ) 6 = 0. Then the E 1

separatrices also cross transversely when λ = λ 0 is fixed and the relative damping parameter τ increases through τ 0 < 1.

Since values τ near τ 0 are in the singular perturbation regime, we obtain het- eroclinic connections E 2 E 1+ and E 1

E 0 , on regular pieces of the slow manifold S 0 , and S

. Indeed, the singular points S 0

1 and S 0

2 on the slow mani- fold S ₀ do not interfere with these slow heteroclinics, as we postpone showing. See (iii) below. By transverse crossing of the E ₁

separatrices, as τ increases through τ ₀ , and by singular perturbations, we also obtain heteroclinics E ₂ E ₁

and E ₁₊ E ₀ , for τ on one side of τ ₀ , say “below”. On the other side, these latter two connections do not exist. See Fig. 5.1.a,b for illustrations of the broken saddle- saddle connection E ₁₊ E ₁

. In either case, the dynamics is Morse-Smale.

By geometric singular perturbation theory as in section 1, we can extend the transverse crossing of the E ₁

separatrices from α = ∞ down to α ≥ α ₀ . More specifically, the transverse crossing occurs at a value λ = λ(τ, α) with λ(τ, ∞ ) = λ(τ), in the above notation for α = + ∞ . Continuity of λ(τ, α), ∂

λ(τ, α) holds up to, and including, α = + ∞ . Note that λ(τ 0 , ∞ ) = λ 0 and ∂

λ(τ, ∞ ) 6 = 0. Solving λ(τ, α) = λ 0 for τ by the implicit function theorem therefore provides a function

τ = τ (α), (5.2)

such that a saddle-saddle separatrix crossing E 1+ E 1

occurs at λ = λ 0 , for any given α ≥ α ₀ . With τ(α) replacing τ ₀ = τ( ∞ ), the claims of the previous paragraph remain valid. In particular, the geometry of Morse-Smale type het- eroclinic connections changes as τ increases through the non-Morse-Smale value τ = τ(α).

To conclude, we now invoke proposition 2.2. By transitivity of the relation in

Morse-Smale systems, E ₂ E ₀ holds for τ with E ₂ E ₁₊ and E ₁₊ E ₀ ; these

were the τ “below” τ(α). In that case, E 2 ∈ ∂B(E 0 ). On the other side of τ(α), in contrast, E 2 cannot connect to E 0 , heteroclinically. Rather, ∂B(E 0 ) = W

(E 1

), which shields E 2 away. In particular, we have proved that the basin boundary

∂B(E 0 ) jumps at τ = τ(α), modulo the three postponed details (i)–(iii).

E

E

S

E

E

u

u

E

S

E

E

E

u

u

Case a) Case b)

B(E )

B(E )

Figure 5.1.

Breaking a singular saddle-saddle heteroclinic, τ ≈ 1. For λ < 1 see case a), for λ > 1 case b).

Shaded: basin B(E

). Drawing is not to scale.

We now give these details. To compute, (i), all equilibria with their Morse indices we fix τ = 0, α = 1, λ = 1. Solving ˙ u 2 = 0 for u 1 and inserting into

˙

u 1 = 0 we obtain a polynomial in u 2 of order six. “Guessing” the two exact solutions

E ₁₊ = (0, 1),

E 1

= ( − 2, − 1) (5.3)

reduces the polynomial to order four. Evaluating explicitly, by symbolic compu- tation, provides two additional real roots. Evaluating the symbolic expressions numerically, we find the remaining two solutions

E ₂ = ( − 3.26, 2.37)

E 0 = ( − 1.75, − 0.91), (5.4)

to two decimals. Computing the Morse indices is trivial.

The second computational detail, (ii), concerns the singular points S 0

1 and S 0

2 on the slow manifold S 0 . By (4.10), (4.13), these points satisfy

f 1 (u 1 ) + f 2 (u 2 ) = 0

f ₁

(u ₁ ) + f ₂

(u ₂ ) = 0 (5.5) Since f ₂

(u ₂ ) = − 2u ₂ , we can eliminate u ₂ from the second equation. The first equation then becomes a quartic polynomial in u ₁

f ₁ (u ₁ ) − ( 1

2 f ₁

(u ₁ )) ² = 0. (5.6)

Evaluating explicitly, by symbolic computation, provides two real solutions. Eval- uating the symbolic expressions numerically, we find

S ₀

1 = ( − 2.46, − 1.35)

S 0

2 = (0.21, − 0.85) (5.7)

to two decimals.

The third computational detail, (iii), concerns the transverse crossing of sepa- ratrices of E ₁

, in the singular limit τ = 1 alias ε = 0, as the coupling constant λ increases through λ = 1. The slow manifold S ₀ does not depend on λ and is given explicitly by (4.10) to be

0 = − f 1 − f 2 = u ² ₂ − 1 + 1

2 ((u 1 + 1) ³ − (u 1 + 1)), (5.8) a very classical algebraic curve which, of course, contains all equilibria. The slow- fast decomposition u = v + w is given explicitly by

A ₁ =

− 1 1 1 − 1

, v = 1

2 (u ₁ + u ₂ ) · 1

1

, w = 1

2 (u ₁ − u ₂ ) · 1

− 1

(5.9) Consequently, the fast system (4.8) leaves the lines

u 1 − u 2 = a (5.10)

invariant, due to w

= 0. The sign of − (f ₁ + f ₂ ) determines the direction of fast motion, as indicated in fig. 5.1. Similarly, the direction of the slow flow (4.9) on S ₀ can be computed easily. Note the reversal of the flow direction on S ₀ at the nondegenerate equilibria and at the singular points S ₀

1 and S ₀

2 where S ₀ is tangent to the fast direction e = (1, 1). By the ordering of w − components of the equilibria and singular points, there exist heteroclinic connections E 2 E 1+ and E 1

E 0 in the regular pieces of the slow manifold S 0 .

The components a

= a

(λ) of the saddles E 1

= (u 1

, u 2

) in the slow w direction are given by a

= u 1

− u 2

. An explicit calculation using the implicit function theorem yields the derivatives at λ = 1:

a

₊ (1) = +1

a

(1) = − 1 (5.11)

Because a

₊ (1) 6 = a

(1), the two fast separatrices (5.10) of E 1

cross transversely with respect to λ, in the singular limit τ % 1.

To complete the proof it remains to show that the fast line (5.10) through the saddles E ₁

,

u ₁ − u ₂ = a = − 1 (5.12)

does not intersect the slow manifold S 0 between those saddles, for λ = 1. Inserting

u 2 = u 1 + 1 in the defining equation (5.8) of S 0 , we indeed obtain a cubic polyno-

mial in u ₁ with the three real roots u ₁ = − 3, − 2, 0. This proves the existence of a

saddle-saddle connection E ₁₊ E ₁

for λ = 1 in the singular limit τ % 1. The

separatrices cross transversely with respect to λ, and the proof of theorem 1.1 is

complete.

6. Discussion

In addition to technical remarks concerning our result, we comment on possible generalizations and the applied question of basin design.

One technical point concerns the saddle-saddle heteroclinic E ₁₊ E ₁

which is responsible for the jump in the basin boundary ∂B(E ₀ ). For large damping α and relative damping τ < 1 near the pure neighbor limit τ = 1, the heteroclinic occurs at coupling constants

λ = λ(τ, α) (6.1)

Rather than invoking analyticity, it should be possible to compute expansions for this jumping surface of ∂B(E 0 ) in parameter space (α, τ, λ) by a Melnikov based method. Some difficulty is posed by the singular limit τ % 1, alias ε & 0.

Periodic forcing of the coupled oscillators by an external vibration of period 2π, by the way, would lead to the additional complication of a rapid forcing of period 2π/α in (1.10). For exponentially small splitting of separatrices with accordingly complicated basin boundaries in such situations see [FS96] and the references there.

For λ = λ ₀ , our basin jump occurs at τ = τ(α) near one: the basin B = B(E ₀ ) develops a tongue which reaches through all the way to the repeller E ₂ ; see Fig.

5.1,b). This tongue is very sharp and thin; in fact it follows the heteroclinic E ₂ E ₁

in the slow manifold S

to within a distance of exponentially small order

O (exp( − c/ε)), (6.2)

uniformly in regions bounded away from E ₁₊ . Although the singular limit fa- cilitates analysis, it may therefore be virtually impossible to detect this tongue by mechanical (or even by numerical) experiments. Nevertheless, we expect the heteroclinic surface (6.1) to extend globally, reaching well into moderate regions of τ , where the jump in the basin boundary becomes quite visible. For a nu- merical phase portrait at τ = 0, λ = 1, α = + ∞ , see Fig. 6.1. By the Sturm and Morse-Smale property, section 3, the surface (6.1) in three parameter space (α, τ, λ) cannot extend down to τ = 0 and hence cannot be globally parametrized over τ, of course.

Saddle-saddle connections E 1+ E 1

in the singular limit τ = 1 may also run through one or several singular points of the slow manifold S 0 . For the simplest case, see Fig. 6.2. In that case, it is the λ-dependent vanishing of the difference of the w-components of E 1+ and of the singular point S 0

1 which determines transverse crossing of the separatrices with respect to the parameter λ. In other geometric situations, it may also be the w-components of two singular points S ₀

1

and S ₀

2 which cross each other transversely with respect to λ and thus cause saddle-saddle heteroclinics and jumping basin boundaries.

For N > 2 degrees of freedom, the singular perturbation geometry in R

becomes more intricate. The slow manifold S ₀ still has codimension 1. The pro-

jection Q onto w along the fast v-direction allows us a superimposed view of the

E

E

E

E

u

u

B(E )

Figure 6.1.

A numerical phase portrait of (1.10), (1.14) with α = + ∞ , τ = 0, λ = 1.

Figure 6.2.

A saddle-saddle heteroclinic orbit running through a singular point of the slow manifold S

.

slow flows (4.9) in v-adjacent regular pieces

v = ψ

(w) (6.3)

of the slow manifold S 0 ; see also (4.12). For example, suppose the ψ + sheet is repelling whereas ψ

attracts in the fast v-direction. Consider equilibria E

of Morse index i(E

) = k, in the respective sheets. By proposition 4.1, the Morse indices i

of E

, viewed as equilibria of the slow flows are given by

i

(E

+ ) = k − 1,

i

(E

) = k (6.4)

In w-coordinates, the superimposed local submanifolds W

(E

+ ) ∩ S and W

(E

) ∩ S can therefore cross transversely, at a distinguished parameter value λ = λ 0 ∈ R . Indeed,

dim(W

(E

+ ) ∩ S) + dim(W

(E

) ∩ S) =

= i

(E

+ ) + (dim S − i

(E

)) = (6.5)

= dim S − 1

add up right. In the singular limit τ = 1, we therefore can have a λ-transverse heteroclinic E

+ E

at λ = λ ₀ .

In theorem 1.1 we have investigated the simplest variant N = 2, dim S = 1, k = 1 of this general construction. It was our principal goal to establish the effect of jumping basins of attraction, as induced by relative changes of damping, even in this most elementary case. The planar effects described, including λ-transverse heteroclinics via one or several singular points, can of course be expected to also occur for N > 2 degrees of freedom.

We have noted the Sturm property (3.4) at purely local damping τ = 0, by which hyperbolicity of equilibria implies the system to be Morse-Smale (proposi- tion 3.1), and therefore structurally stable. In absence of degenerate heteroclinics, the Morse-Smale property also holds for pure neighbor damping τ = 1. In the dissipative case, a detailed enumeration of the connection patterns of heteroclinics E E

is available for the Sturm case τ = 0; see [FR96a] and the references there. For τ = 1, an analogous classification is missing. For example, it is not clear whether or not Morse-Smale attractors can arise, for τ near one, which are not conjugate to any Morse-Smale attractor of Sturm type as classified in [FR96a].

A related question, for all τ, is the design of basins of attraction by suitable choices of damping A(τ) and coupling L. We have seen how basin boundaries can change drastically by adjusting A(τ) alone, even in the planar case N = 2. Optimality issues arise, naturally, when strongly damped systems of coupled mechanical oscillators are considered in practical applications. For example, one might want to enlarge the basin B(E 0 ) of a desirable stable equilibrium state E 0 . With our paper, we hope to have achieved a first small step in such directions.

Acknowledgment

This work was supported by DFG and GTZ, “Nonlinear Vibrations of Coupled

Oscillators.” Hospitality of the University Hassan II, Casablanca and the Free

University, Berlin, is mutually acknowledged. For efficient and diligent typesetting

we are much indebted to Mrs. Andrea Behm and Mrs. Monika Schmidt.

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Bernold Fiedler Freie Universit¨ at Berlin Arnimallee 2-6, D-14195 Berlin GERMANY

Mohamed Belhaq and Mohamed Houssni Faculty of Sciences Ain Chock

University Hassan II BP5366 Maarif, Casablanca MOROCCO

(Received: June 11, 1997; revised: November 4, 1997)